Theorem xmstopn 24393

Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	⊢ 𝐽 = (TopOpen‘𝐾)
isms.x	⊢ 𝑋 = (Base‘𝐾)
isms.d	⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
xmstopn	⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 24389	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	4	simprbi 496	1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   × cxp 5620   ↾ cres 5624  ‘cfv 6490  Basecbs 17134  distcds 17184  TopOpenctopn 17339  MetOpencmopn 21297  TopSpctps 22874  ∞MetSpcxms 24259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-res 5634  df-iota 6446  df-fv 6498  df-xms 24262

This theorem is referenced by:  imasf1oxms  24431  ressxms  24467  prdsxmslem2  24471  tmsxpsmopn  24479  xmsusp  24511  cmetcusp1  25307  minveclem4a  25384  minveclem4  25386  qqhcn  34097  rrhcn  34103  rrexthaus  34113  dya2icoseg2  34384  sitmcl  34457